Q114 / 160 · AP EAMCET 2023 (17 May Shift 1) · Mathematics

If $a \in \mathbb{Z}^{+},[x]$ is greatest integer not more than $x$ and $\int_0^a 2^{[x]} d x=127$, then $a=$

  • A
    $6$
  • B
    $7$
  • Correct answer
    C
    $8$
  • D
    $9$

Solution

$\begin{aligned} & \text {} \int_0^a 2^{[x]} d x \\ & =\int_0^1 2^0 d x+\int_1^2 2^1 d x+\int_2^3 2^2 d x+\ldots \int_{a-1}^a 2^{a-1} d x=127 \\ & \Rightarrow 2^0+2^1+2^2+\ldots+2^{a-1}=127 \\ & \Rightarrow \frac{1\left[2^a-1\right]}{2-1}=127 \Rightarrow 2^a=128 \Rightarrow 2^a=2^8 \Rightarrow a=8\end{aligned}$

Hence, the correct answer is option: 'C'.

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